<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Olive" title="b1">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Olive" title="b1">f1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c2">f1</font></span><br/>



<a NAME="E1:10"/><span class="lab"><font color="Green" title="E10">F</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_2.html#D39" title="VALUED_2:def.39">VALUED_2:def 39</a></span>;<br/>
<a NAME="E2:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c2">f1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_4.html#D3" target="_self" title="NFCONT_4:def.3">Def5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:10_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><a NAME="E2:10_1"/><span class="lab"><font color="Green" title="E13">G</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E10">F</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="nfcont_4.html#D3" target="_self" title="NFCONT_4:def.3">Def5</a></span>;<br/><a NAME="E4:10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_2.html#K43" title="VALUED_2:func.43">[#]</a> <span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="valued_2.html#D39" title="VALUED_2:def.39">VALUED_2:def 39</a></span>;<br/><span class="kw">hence </span><a NAME="E5:10_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E13">G</font></span></a>, <a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:10"/>
<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c2">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
